Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables
Jean-Luc Marichal
Applied Mathematics Unit, University of Luxembourg 162A, avenue de la Fa¨ıencerie, L-1511 Luxembourg, Luxembourg
jean-luc.marichal[at]uni.lu
Abstract
By using some basic calculus of multiple inte- gration, we provide an alternative expression of the integral
Z
]a,b[n
f(x,minxi,maxxi)dx, in which the minimum and the maximum are replaced with two single variables. We demonstrate the usefulness of that expression in the computation of orness and andness average values of certain aggregation func- tions. By generalizing our result to Riemann- Stieltjes integrals, we also provide a method for the calculation of certain expected values and distribution functions.
Keywords: Multivariate integration, aggre- gation function, Cauchy mean, distribution function, expected value, andness, orness.
1 Introduction
Let a, b ∈ R∪ {−∞,+∞}, with a < b, and consider an integral over ]a, b[n whose integrand displays an ex- plicit dependence on the minimum and/or the maxi- mum of the variables, that is, an integral of the form
Z
]a,b[n
f(x,minxi,maxxi)dx. (1) In this note we provide an alternative expression of this integral, in which the minimum and the maxi- mum are replaced with two single variables. When the integral is tractable, that alternative expression generally makes the integral much easier to evaluate.
For instance, when the integrand depends only on the minimum and the maximum of the variables, we ob-
tain the following identity Z
]a,b[n
f(minxi,maxxi)dx
=n(n−1) Z b
a
dv Z v
a
f(u, v)(v−u)n−2du, (2) and hence, for certain functions f, the integral be- comes very easy to evaluate.
The alternative expression we present for integral (1) is given in the next section (see Theorem 2.1). The method we employ to obtain that expression merely consists in dividing the domain ]a, b[nintonpolyhedra chosen in such a way that the minimum and maximum functions simply become single variables.
This method can be very efficient in the evaluation of many integrals that would normally require difficult and tedious computations. As an example, consider the variance of a samplex∈[a, b]n from a given pop- ulation, namely
s2(x) = 1 n−1
Xn
i=1
³ xi−1
n Xn
j=1
xj
´2 .
The average value over [a, b]n of the variance-to-range ratio function can be easily calculated by using our method. We merely obtain
1 (b−a)n
Z
[a,b]n
s2(x)
maxxi−minxidx=n+ 2
12n (b−a).
(3) This note is set out as follows. In Section 2 we state and prove the main result. In Section 3 we provide an application of our result to internal functions, also called Cauchy means, which can be classified accord- ing to their location within the range of the variables.
A similar application to conjunctive and disjunctive functions is also investigated. In Section 4 we show
how the direct generalization of our result to Riemann- Stieltjes integrals enables us to consider the evaluation of certain expected values from various distributions.
We will use the following notation throughout. For any n-tuplex, we denote by (x| xj =u) then-tuple whose ith coordinate is u ifi = j, and xi otherwise.
Also, for any integer n>1, we set [n] :={1, . . . , n}.
2 Main result
In this section we present our main result which con- sists of an alternative expression of integral (1). We start with a preliminary lemma, which concerns the particular cases of functions involving either the min- imum or the maximum of the variables.
Lemma 2.1. Let f : ]a, b[n+1 → R be an integrable function. Then we have
Z
]a,b[n
f(x,minxi)dx
= Xn
j=1
Z b
a
du Z
]u,b[n−1
f(x, u|xj=u) Y
i∈[n]\{j}
dxi, Z
]a,b[n
f(x,maxxi)dx
= Xn
j=1
Z b
a
dv Z
]a,v[n−1
f(x, v|xj=v) Y
i∈[n]\{j}
dxi.
Proof. Consider the following n-dimensional open polyhedra
Pj:={x∈]a, b[n: xi> xj∀i6=j} (j ∈[n]).
They are pairwise disjoint. Indeed, if x ∈ Pj ∩Pk, with j 6= k, then xk > xj and xj > xk, which is a contradiction. Moreover, the union of their set closures covers ]a, b[n. Indeed, for anyx∈]a, b[nthere is always j∈[n] such thatxi>xj for alli6=j.
Therefore, for any integrable function f : ]a, b[n+1 → R, we have
Z
]a,b[n
f(x,minxi)dx
= Xn
j=1
Z
Pj
f(x,minxi)dx
= Xn
j=1
Z b
a
dxj
Z
]xj,b[n−1
f(x, xj) Y
i∈[n]\{j}
dxi, which proves the first formula. The second formula can be established similarly by considering the polyhedra
Qj :={x∈]a, b[n: xi< xj∀i6=j} (j∈[n]).
Lemma 2.1 is interesting in its own right since it pro- vides special cases of the main result. For instance, by applying the first formula, we immediately obtain the following identity, which will be used in the next section (see Example 3.2). For anyS⊆[n], we have
Z
]a,b[n
f¡ mini∈S xi
¢dx
= (b−a)n−|S||S|
Z b
a
f(u)(b−u)|S|−1du. (4) Let us now state our main result, which follows imme- diately from two applications of Lemma 2.1.
Theorem 2.1. Let n>2 and letf : ]a, b[n+2→Rbe an integrable function. Then we have
Z
]a,b[n
f(x,minxi,maxxi)dx
= Xn
j,k=1 j6=k
Z b
a
dv Z v
a
du× Z
]u,v[n−2
f(x, u, v|xj=u, xk=v) Y
i∈[n]\{j,k}
dxi.
A direct use of this result leads to formula (3). In- deed, as the integrand is symmetric in its variables, we simply need to consider
f(x, u, v|xj=u, xk =v)
= 1
v−us2(x1, . . . , xn−2, u, v), (5) where, for any fixed a < u < v < b, the right-hand side is a quadratic polynomial inx1, . . . , xn−2.
3 Application to aggregation function theory
We now apply our main result to the computation of orness and andness average values of internal functions and to the computation of idempotency average values of conjunctive and disjunctive functions.
3.1 Internal functions
We recall the concept of internal functions, which was introduced in the theory of means and aggregation functions.
Definition 3.1. A functionF : ]a, b[n→Ris said to beinternal if
minxi6F(x)6maxxi (x∈]a, b[n).
Internality is a property introduced by Cauchy [3]
who considered in 1821 the mean of n independent variablesx1, . . . , xn as a functionF(x1, . . . , xn) which should be internal to the set ofxivalues. Internal func- tions, also called Cauchy means, are very often encoun- tered in the literature on aggregation functions. Most of the classical means, such as the arithmetic mean, the geometric mean, and their weighted versions, are Cauchy means. For an overview on means and aggre- gation functions, see the edited book [2].
It is straightforward to see that a functionF: ]a, b[n→ R is internal if and only if there is a functionf from ]a, b[n\diag(]a, b[n) to [0,1] such that
F(x) = minxi+f(x) (maxxi−minxi), where diag(]a, b[n) :={(x, . . . , x)∈]a, b[n:x∈]a, b[}.
Starting from this observation, Dujmovi´c [4] (see also [6]) introduced the following concepts of local orness and andness functions, rediscovered independently by Fern´andez Salido and Murakami [8] as orness and and- ness distribution functions.
Definition 3.2. The orness distribution function (resp. andness distribution function) associated with an internal functionF : ]a, b[n→Ris a functionodfF
(resp.adfF), from ]a, b[n\diag(]a, b[n) to [0,1], defined as
odfF(x) = F(x)−minxi
maxxi−minxi
(resp.adfF(x) = maxxi−F(x) maxxi−minxi).
Thus defined, the orness distribution function (resp.
andness distribution function) associated with an in- ternal function F : ]a, b[n → R measures, at each x∈]a, b[n, the extent to whichF(x) is close to maxxi
(resp. minxi), that is, the extent to whichF(x) has a disjunctive (resp. conjunctive) or orlike (resp. andlike) behavior.
To measure the average orness or andness quality of an internal function over its domain, Dujmovi´c [4] also introduced the concepts of mean local orness and and- ness, later called orness and andness average values by Fern´andez Salido and Murakami [8].
Definition 3.3. Theorness average value (resp.and- ness average value) of an internal and integrable func- tionF : ]a, b[n →Ris defined as
odfF = 1 (b−a)n
Z
]a,b[n
odfF(x)dx (resp.adfF = 1
(b−a)n Z
]a,b[n
adfF(x)dx).
As an immediate property, we note that odfF(x) +adfF(x) = 1,
which entails odfF +adfF = 1. Thus, as expected, bothodfF andadfF render the same information and hence we can restrict ourselves to the computation of odfF.
Even though the computation of odfF remains very difficult in most of the cases, Theorem 2.1 enables us to rewrite this integral in a more practical form, namely odfF
= 1
(b−a)n Xn
j,k=1 j6=k
Z b
a
dv Z v
a
du×
Z
]u,v[n−2
F(x|xj=u, xk=v)−u v−u
Y
i∈[n]\{j,k}
dxi.
The following two examples demonstrate the power of this formula:
Example 3.1. The orness average value over [0,1]n of the geometric mean
G(n)(x) = Yn
i=1
x1/ni
is given forn= 2 by
odfG(2)= ln 4−1 and forn >2 by
odfG(n)
= n(n−1)³ n n+ 1
´n−2Z 1
0
xn(1−xn+1)n−2 1−xn dx
− 1 n−2.
Example 3.2. Consider a function of the form Ca(n)(x) = X
S⊆[n]
a(S) min
i∈Sxi,
where the set functiona: 2[n]→Rfulfills a(∅) = 0 and X
S⊆[n]
a(S) = 1
and is chosen so that the functionCa(n)is nondecreas- ing in each variable. Such a function is known in aggregation function theory as a Lov´asz extension or a discrete Choquet integral (see for instance [9, 11]).
As particular cases, we can consider weighted means
P
iwixi and convex combinations P
iwix(i) of order statistics, also known as “ordered weighted averaging”
functions (see [13]).
The orness average value over [0,1]n of Ca(n) is given by
odfC(n)
a = 1
n−1 X
S⊆[n]
a(S)n− |S|
|S|+ 1.
To overcome the difficulty of calculating intractable or- ness average values, Dujmovi´c [5] introduced the next concept of global orness and andness measures (see also [6, 8]). Denote by F the average value of any in- ternal and integrable functionF : ]a, b[n →Rover its domain, that is,
F := 1
(b−a)n Z
]a,b[n
F(x)dx.
Definition 3.4. Theglobal orness value(resp.global andness value) of an internal and integrable function F : ]a, b[n→Ris defined as
ornessF = F−Min Max−Min (resp. andnessF = Max−F
Max−Min),
where Min and Max are, respectively, the minimum and maximum functions defined in ]a, b[n.
For example, considering the geometric mean G(n)(x) =Qn
i=1x1/ni in [0,1]n, we simply obtain ornessG(n) =− 1
n−1+n+ 1 n−1
³ n n+ 1
´n
. Considering the discrete Choquet integral Ca(n) in [0,1]n, as defined in Example 3.2, we get
ornessC(n)
a = 1
n−1 X
S⊆[n]
a(S)n− |S|
|S|+ 1. Surprisingly enough, in [0,1]n we have
ornessC(n)
a =odfC(n)
a ,
that is, for any discrete Choquet integral, the global orness value identifies with the orness average value, a result already reached by Fern´andez Salido and Mu- rakami [8] for the special case of symmetric Choquet integrals, that is, convex combinations of order statis- tics.
The interesting question of determining those inter- nal functions F : ]a, b[n → R fulfilling the equation ornessF =odfF remains open.
3.2 Conjunctive and disjunctive functions Let us now consider conjunctive and disjunctive func- tions.
Definition 3.5. A functionF : ]a, b[n→Ris said to beconjunctive (resp.disjunctive) if
a6F(x)6minxi
¡resp. maxxi6F(x)6b¢ . Prominent examples of conjunctive (resp. disjunc- tive) functions in the literature are t-norms (resp. t- conorms), which are symmetric, associative, and non- decreasing functions, from [0,1]2to [0,1], with 0 (resp.
1) as the neutral element. For an account on t-norms and t-conorms, see for instance the book by Alsina et al. [1].
Clearly, a function F : ]a, b[n → R is conjunctive (resp. disjunctive) if and only if there is a function f : ]a, b[n→[0,1] such that
F(x) =a+f(x)(minxi−a)
¡resp.F(x) =b−f(x)(b−maxxi)¢ .
Just as for the orness and andness distribution func- tions, we can naturally define the concept of idempo- tency distribution function associated with a conjunc- tive (resp. disjunctive) function F : ]a, b[n → R as a measure, at eachx∈]a, b[n, of the extent to whichF is idempotent (i.e., such thatF(x, . . . , x) =x), that is, the extent to whichF is close to minxi (resp. maxxi).
Definition 3.6. The idempotency distribution func- tion associated with a conjunctive (resp. disjunctive) functionF : ]a, b[n →Ris a function idfF : ]a, b[n → [0,1], defined as
idfF(x) = F(x)−a minxi−a (resp.idfF(x) = b−F(x)
b−maxxi).
We can now introduce the concept of idempotency av- erage value as follows.
Definition 3.7. The idempotency average value of a conjunctive or disjunctive function F : ]a, b[n → Ris defined as
idfF = 1 (b−a)n
Z
]a,b[n
idfF(x)dx.
According to Lemma 2.1, for any conjunctive function
F : ]a, b[n→Rfor instance, we can write
idfF = 1
(b−a)n Xn
j=1
Z b
a
du× Z
]u,b[n−1
F(x|xj =u)−a u−a
Y
i∈[n]\{j}
dxi.
The following concept of global idempotency value was introduced by Koles´arov´a [10] for t-norms as an idem- potency measure:
Definition 3.8. The global idempotency value of a conjunctive (resp. disjunctive) functionF: ]a, b[n→R is defined by
idempF = F−a
Min−a (resp. idempF = b−F b−Max).
Example 3.3. The idempotency average value and the global idempotency value over [0,1]n of the prod- uct
P(n)(x) = Yn
i=1
xi,
which is a conjunctive function, is given by idfP(n)= 2n−1
¡2n−1
n
¢
while its global idempotency value is given by idempP(n)= n+ 1
2n .
4 Application to probability theory
Since the idea behind our results merely consists in breaking the integration domain into smaller regions, Lemma 2.1 and Theorem 2.1 can be straightforwardly extended to Riemann-Stieltjes integrals, thus making it possible to consider average values from various probability distributions.
Consider a measurable functiong : Rn+2 →Rand n independent random variablesX1, . . . , Xn,Xi(i∈[n]) having distribution functionFi(x). Define the random variableYg as
Yg :=g(X,minXi,maxXi), whereXdenotes the vector (X1, . . . , Xn).
The direct generalization of Theorem 2.1 to Riemann- Stieltjes integrals can be used to evaluate the expected value ofYg, namely
E[Yg] = Z
Rn
g(x,minxi,maxxi)dF1(x1)· · ·dFn(xn).
It can also be used in the evaluation of the distribution function ofYg, which is defined as
Fg(z) = E[H(z−Yg)]
= Z
Rn
H¡
z−g(x,minxi,maxxi)¢
× dF1(x1)· · ·dFn(xn),
where H : R→ {0,1} is the Heaviside step function, defined byH(x) = 1 if x>0, and 0 otherwise. Note that the case where Yg is a lattice polynomial (max- min combination) of the variablesX1, . . . , Xnhas been thoroughly investigated by the author in [12].
To keep our exposition simple, let us examine the spe- cial case whereYgdepends only on minXiand maxXi, that is,
Yg:=g(minXi,maxXi),
where g : R2 → R is a measurable function. In this case, our method immediately leads to
E[Yg] = Xn
j,k=1 j6=k
Z ∞
−∞
dFk(v)× Z v
−∞
g(u, v)dFj(u) Z
]u,v[n−2
Y
i∈[n]\{j,k}
dFi(xi)
= Xn
j,k=1 j6=k
Z ∞
−∞
dFk(v)× Z v
−∞
g(u, v) Y
i∈[n]\{j,k}
¡Fi(v)−Fi(u)¢ dFj(u).
In the particular case where the random vari- ables X1, . . . , Xn are independent and identically dis- tributed, each with distribution functionF(x), the ex- pected value clearly reduces to
E[Yg] =n(n−1) Z ∞
−∞
dF(v)×
Z v
−∞
g(u, v)¡
F(v)−F(u)¢n−2
dF(u), (6) which generalizes formula (2).
Example 4.1. For exponential variablesX1, . . . , Xn, each with distribution functionF(x) = 1−e−λx (x >
0), we simply have E[Yg] =n(n−1)
Z ∞
0
λ e−λvdv×
Z v
0
g(u, v)¡
e−λu−e−λv¢n−2
λ e−λudu.
Using the change of variablesx=e−λuandy=e−λu− e−λv, this integral can be easily rewritten as
E[Yg] =n(n−1) Z 1
0
dx×
Z x
0
yn−2g¡
−1
λ ln(x),−1
λ ln(x−y)¢ dy.
Example 4.2. Let us calculate the distribution func- tion and the raw moments of the random variable
Y =maxXi−minXi
maxXi
from the uniform distribution over ]0,1]n.
The raw moments can be calculated very easily from (6). For any integer r>0, we have
E[Yr] =n(n−1) Z 1
0
dv×
Z v
0
³v−u v
´r
(v−u)n−2du= n−1 n+r−1. On the other hand, the distribution of Y is simply given by
F(z) =n(n−1) Z 1
0
dv Z v
0
H¡
z−v−u v
¢(v−u)n−2du,
that is,
F(z) =
0, ifz60, zn−1, if 06z61, 1, if 16z.
References
[1] C. Alsina, M. J. Frank, and B. Schweizer. Asso- ciative functions – Triangular norms and copulas.
World Scientific Publishing Co. Pte. Ltd., Hack- ensack, NJ, 2006.
[2] T. Calvo, G. Mayor, and R. Mesiar, editors. Ag- gregation operators: new trends and applications, volume 97 ofStudies in Fuzziness and Soft Com- puting. Physica-Verlag, Heidelberg, 2002. ISBN 3-7908-1468-7.
[3] A. L. Cauchy. Cours d’analyse de l’Ecole Royale Polytechnique, Vol. I. Analyse alg´ebrique. De- bure, Paris, 1821.
[4] J. J. Dujmovi´c. A generalization of some func- tions in continuous mathematical logic – Eval- uation function and its applications, (in Serbo- Croatian). InProc. Informatica Conference, Bled, Yugoslavia, 1973. Paper d27.
[5] J. J. Dujmovi´c. Weighted conjunctive and dis- junctive means and their application in system evaluation. J. Univ. Belgrade, EE Dept., Series Mathematics and Physics, (483):147–158, 1974.
[6] J. J. Dujmovi´c. Seven flavors of andness/orness.
In B. De Baets, J. Fodor, and D. Radojevi´c, edi- tors,Proc. Eurofuse 2005, pages 81–92, Institute
“MihajloPupin”, Belgrade, 2005. ISBN 86-7172- 022-5.
[7] B. Eisenberg and R. Sullivan. Crofton’s differen- tial equation. Am. Math. Mon., 107(2):129–139, 2000.
[8] J. M. Fern´andez Salido and S. Murakami. Extend- ing Yager’s orness concept for the OWA aggrega- tors to other mean operators. Fuzzy Sets Syst., 139(3):515–542, 2003.
[9] M. Grabisch, J.-L. Marichal, and M. Roubens.
Equivalent representations of set functions.Math.
Oper. Res., 25(2):157–178, 2000.
[10] A. Koles´arov´a. Revision of parametric evaluation of aggregation functions. In Proc. 13th Zittau Fuzzy Colloquium, pages 202–211, Zittau, Ger- many, September 2006.
[11] J.-L. Marichal. An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Trans. Fuzzy Syst., 8(6):800–807, 2000.
[12] J.-L. Marichal. Cumulative distribution functions and moments of lattice polynomials. Statistics &
Probability Letters, 76(12):1273–1279, 2006.
[13] R. R. Yager and J. Kacprzyk, editors. The Or- dered Weighted Averaging operators. Kluwer Aca- demic Publishers, USA, 1997.
